Principal ideal problem and ideal shortest vector over rational primes in power-of-two cyclotomic fields

The shortest vector problem (SVP) over ideal lattices is closely related to the Ring-LWE problem, which is widely used to build post-quantum cryptosystems. Power-of-two cyclotomic fields are frequently adopted to instantiate Ring-LWE. Pan et al. (EUROCRYPT~2021) explored the SVP over ideal lattices via the decomposition fields and, in particular determined the length of the shortest vector in prime ideals lying over rational primes $p\equiv3,5\pmod{8}$ in power-of-two cyclotomic fields via explicit construction of reduced lattice bases. In this work, we first provide a new method (different from analyzing lattice bases) to analyze the length of the shortest vector in prime ideals in $\mathbb{Z}[\zeta_{2^{n+1}}]$ when $p\equiv3,5\pmod{8}$. Then we precisely characterize the length of the shortest vector in the cases of $p\equiv7,9\pmod{16}$. Furthermore, we derive a new upper bound $\sqrt[4]{2^{2n+1}p}$ for this length, which is tighter than the bound $2^n\sqrt[4]{p}$ obtained from Minkowski's theorem. Our key technique is to investigate whether a generator of a principal ideal can achieve the shortest length after embedding as a vector. If this holds for the ideal, finding the shortest vector in this ideal can be reduced to finding its shortest generator.